Skip to main content
SpringerLink
Log in

Diameter and Convex Hull of Points Using Space Subdivision in E2 and E3

  • Conference paper
  • First Online:
Computational Science and Its Applications – ICCSA 2020 (ICCSA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12249))

Included in the following conference series:

Abstract

Convex hull of points and its diameter computation is a frequent task in many engineering problems, However, in engineering solutions, the asymptotic computational complexity is less important than the computational complexity for the expected data size to be processed. This contribution describes “an engineering solution" of the convex hulls and their diameter computation using space-subdivision and data-reduction approaches. This approach proved a significant speed-up of computation with simplicity of implementation. Surprisingly, the experiments proved, that in the case of the space subdivision the reduction of points is so efficient, that the “brute force" algorithms for the convex hull and its diameter computation of the remaining points have nearly no influence to the time of computation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Andrew, A.: Another efficient algorithm for convex hulls in two dimensions. Inf. Process. Lett. 9(5), 216–219 (1979)

    Article  Google Scholar 

  2. Bradford, B., Dobkin, D., Dobkin, H., Huhdanpaa, D.P.: The quickhull algorithm for convex hulls. ACM Trans. Mathe. Softw. 22(4), 469–483 (1996)

    Article  MathSciNet  Google Scholar 

  3. Bykat, A.: Convex hull of a finite set of points in two dimensions. Inf. Process. Lett. 7(6), 296–298 (1978)

    Article  MathSciNet  Google Scholar 

  4. Chan, T.: Optimal output-sensitive convex hull algorithms in two and three dimensions. Discrete Comput. Geom. 16(4), 361–368 (1996)

    Article  MathSciNet  Google Scholar 

  5. Chand, D., Kapur, S.: An algorithm for convex polytopes. J. ACM 17(1), 78–86 (1970)

    Article  MathSciNet  Google Scholar 

  6. Dobkin, D., Snyder, L.: On a general method for maximizing and minimizing among certain geometric problems. In: FOCS-IEEE Symposium on Foundations of Computer Science Proceedings, pp. 9–17 (1979)

    Google Scholar 

  7. Dévai, F., Szendrényi, T.: Comments on convex hull of a finite set of points in two dimensions. Inf. Process. Lett. 9(3), 141–142 (1979)

    Article  MathSciNet  Google Scholar 

  8. Eddy, W.: Algorithm 523: convex, a new convex hull algorithm for planar sets. ACM Trans. Math. Softw. (TOMS) 3(4), 398–403 (1997)

    Article  MathSciNet  Google Scholar 

  9. Graham, R.: An efficient algorithm for determining the convex hull of a finite planar set. Inf. Process. Lett. 1(4), 132–133 (1972)

    Article  Google Scholar 

  10. Gupta, N., Sen, S.: Faster output-sensitive parallel algorithms for 3D convex hulls and vector maxima. J. Para. Distrib. Comput. 63(4), 488–500 (2003)

    Article  Google Scholar 

  11. Halton, J.: Algorithm 247: radical-inverse quasi-random point sequence. CACM 7(12), 701–702 (1964)

    Article  Google Scholar 

  12. Jarvis, R.: On the identification of the convex hull of a finite set of points in the plane. Inf. Process. Lett. 2(1), 18–21 (1973)

    Article  Google Scholar 

  13. Kallay, M.: The complexity of incremental convex hull algorithms in \({R}^d\). Inf. Process. Lett. 19(4), 197 (1984)

    Article  MathSciNet  Google Scholar 

  14. Kirkpatrick, D., Seidel, R.: The ultimate planar convex hull algorithm? SIAM J. Comput. 15(1), 287–299 (1986)

    Article  MathSciNet  Google Scholar 

  15. Liu, G., Chen, C.: A new algorithm for computing the convex hull of a planar point set. J. Zhejiang Univ. Sci. A 8(8), 1210–1217 (2007)

    Article  Google Scholar 

  16. Löffler, M.: A manual comparison of convex hull algorithms (multimedia exposition). In: 35th International Symposium on Computational Geometry (SoCG 2019), volume 129 of Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, pp. 65:1–65:2. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2019)

    Google Scholar 

  17. Nguyen, K., Song, C., Ryu, J., Thanh, P.A., Hoang, N.-D., Kim, D.-S.: Quickhulldisk: a faster convex hull algorithm for disks. Appl. Math. Comput. 363, 124626 (2019)

    MathSciNet  MATH  Google Scholar 

  18. Preparata, F., Hong, S.: Convex hulls of finite sets of points in two and three dimensions. CACM 20(2), 87–93 (1997)

    Article  MathSciNet  Google Scholar 

  19. Skala, V.: Fast \({O}_{{\rm expected}}({N})\) algorithm for finding exact maximum distance in \({E}^2\) instead of \({O(N}^2)\) or \({O(N lg N)}\). In: ICNAAM 2013, AIP Proceedings, USA, vol. 1558, pp. 2496–2499. AIP Publishing (2013)

    Google Scholar 

  20. Skala, V., Majdisova, Z.: Fast algorithm for finding maximum distance with space subdivision in E\(^2\). In: Zhang, Y.-J. (ed.) ICIG 2015. LNCS, vol. 9218, pp. 261–274. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-21963-9_24

    Chapter  Google Scholar 

  21. Skala, V., Majdisova, Z., Smolik, M.: Space subdivision to speed-up convex hull construction in \({E}^3\). Adv. Softw. Eng. 91(C), 12–22 (2016)

    Google Scholar 

  22. Skala, V., Smolik, M.: Simple and fast \({O}exp{(N)}\) algorithm for finding an exact maximum distance in \({E}^2\) instead of \({O(N}^2)\) or \({O(N lg N)}\). In: Misra, S., et al. (eds.) ICCSA 2019. LNCS, vol. 11619, pp. 367–380. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-24289-3_27

    Chapter  Google Scholar 

  23. Skala, V., Smolik, M., Majdisova, Z.: Reducing the number of points on the convex hull calculation using the polar space subdivision in \({E}^2\). In: 29th SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI 2016), pp. 40–47. IEEE (2016)

    Google Scholar 

  24. Stein, A., Geva, E., El-Sana, J.: Cudahull: fast parallel 3D convex hull on the GPU. Comput. Graph. 36(4), 265–271 (2012)

    Article  Google Scholar 

  25. Sugihara, K.: Robust gift wrapping for the three-dimensional convex hull. J. Comput. Syst. Sci. 49(2), 391–407 (1994)

    Article  MathSciNet  Google Scholar 

  26. Toussaint, J., McAlear, G.T.: A simple o(n log n) algorithm for finding the maximum distance between two finite planar sets. Pattern Recogn. Lett. 1(1), 21–24 (1982)

    Article  Google Scholar 

  27. Xue, J., Li, Y., Janardan, R.: On the expected diameter, width, and complexity of a stochastic convex hull. Comput. Geom. 82, 16–31 (2019)

    Article  MathSciNet  Google Scholar 

  28. Yao, A., Andrew, C.: A lower bound to finding convex hulls. J. ACM 28(4), 780–787 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgments

The author would like to thank colleagues and students at the University of West Bohemia, Pilsen, for their discussions and suggestions, especially to Zuzana Majdisova and Michal Smolik for recent implementations and experiments made, and to anonymous reviewers for their valuable comments and hints provided.

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Faculty of Applied Sciences, University of West Bohemia, 301 00, Pilsen, CZ, Czech Republic

    Vaclav Skala

Authors
  1. Vaclav Skala
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vaclav Skala .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Chair- Center of ICT/ICE, Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Clayton School of Information Technology, Monash University, Clayton, VIC, Australia

    David Taniar

  7. Department of Information Science, Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A.C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

  11. Department of Neurology, University of Massachusetts Medical School, Worcester, MA, USA

    Yeliz Karaca

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Skala, V. (2020). Diameter and Convex Hull of Points Using Space Subdivision in E2 and E3. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2020. ICCSA 2020. Lecture Notes in Computer Science(), vol 12249. Springer, Cham. https://doi.org/10.1007/978-3-030-58799-4_21

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-58799-4_21

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-58798-7

  • Online ISBN: 978-3-030-58799-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation